The Anisotropic Aphid: Three-Dimensional Induction Modeling of Electrical Texture with Mixed Potentials
Abstract:At the macroscopic scale, where the e-folding distance of low-frequency electromagnetic fields in conductive geomaterials is much larger than the size of organized heterogeneities such as fracture sets or laminations that constitute the geologic texture therein, electrical properties can be conveniently approximated by a generalized 3x3 tensor σ. Less convenient, however, are the algorithmic consequences of this approximation in electromagnetic modeling of 3D induction methods for geophysical exploration. Previous efforts at modelling generalized anisotropy with finite differences on a staggered Cartesian grid (e.g. Weiss and Newman, 2002; Wang and Fang, 2001) are posed in terms of the electric field with its governing "curl-curl" equation and well-documented null-space issues at low induction numbers. In contrast, Weiss (2013) proposed an alternate full-physics formulation in terms of Lorenz-gauged magentic vector A and electric scalar Φ potentials (Project APhiD) that eliminates the troublesome curl-curl operator, with ultrabroadband examples drawn from geologies with scalar, isotropic conductivity over the frequency range 10-2-1010 Hz. Here, the anisotropic theory presented in Weiss (2013) is implemented with finite differences on a Cartesian grid.
Briefly stated, in this theoretical approach the conductivity tensor σ is split in terms of a rotationally-invariant isotropic conductivity σ* = ⅓ Tr(σ) and the residual σ - σ*I. This splitting decomposes the resulting finite difference coefficient matrix K into the sum Kiso + Kaniso, where the Kiso term is the coefficient matrix for the isotropic medium σ*, thus enabling reuse of the various routines previously developed for computing matrix coefficients in the isotropic case. Treatment of anisotropy is algorithmically therefore restricted to computing the coefficients in the sparse matrix Kaniso consisting of simple inner products of (σ - σ*I) · (A-∇Φ) and their divergence. In keeping with the philosophy of economizing the compute resource footprint, the anisotropic finite difference algorithm is solved in a matrix-free formulation whereby coefficient matrices are computed as needed at each step of the iterative BiCG-STAB solver used to solve the finite difference system of equations.